An inverse problem for a general convex domain with impedance boundary conditions

Abstract

The spectral function θ ( t ) = ∑ n = 1 ∞ exp ⁡ ( − t λ n ) \theta \left ( t \right ) = \sum \nolimits _{n = 1}^\infty {\exp \left ( { - t{\lambda _n}} \right )} , where { λ n } n = 1 ∞ \left \{ {{\lambda _n}} \right \}_{n = 1}^\infty are the eigenvalues of the Laplace operator Δ = ∑ i = 1 2 ( ∂ / ∂ x i ) 2 \Delta = \sum \nolimits _{i = 1}^2 {{{\left ( {\partial /\partial {x^i}} \right )}^2}} in the x 1 x 2 {x^1}{x^2} -plane, is studied for a general convex domain Ω ⊆ R 2 \Omega \subseteq {R^2} with a smooth boundary ∂ Ω \partial \Omega together with a finite number of piecewise smooth impedance boundary conditions on the parts Γ 1 , . . . , Γ m {\Gamma _{1,...,}}{\Gamma _m} of ∂ Ω \partial \Omega such that ∂ Ω = U j = 1 m Γ j \partial \Omega = U_{j = 1}^m{\Gamma _j} .